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JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact email@example.com. A SCALE FOR MEASURING PUPILS' ABILITY TO DEMONSTRATE GEOMETRICAL THEOREMS J. H. MINNICK University of Pennsylvania It is desirable that a scale shall be developed for the measurement of each of those abilities upon which the mastery of geometry is dependent. Among these is the ability to demonstrate a theorem when the figure is drawn, and the hypothesis and conclusion are given. The chief difficulty in developing a scale for measuring this ability is the time element. The writing of a complete demonstration requires too much of the pupil's time, and the scoring of such a scale is time consuming. On a former occasion 1 , the writer used a geometry test in which the drawing for a theorem was given and the hypothesis and conclusion were stated. The pupil was then asked to make any additional drawings necessary to prove the theorem. Although the exercises of this test were poorly selected, the results were so favorable that it seemed possible* by the use of this method, to construct a satisfactory scale. This scale would have the advantages of requiring no writing on the part of the pupil and of being easily scored. The following is a report of an attempt to construct such a scale. From various sources eleven exercises, apparently suited to the purpose of the scale, were selected. The time available for giving the test made it impossible for each pupil to solve all the exercises. In order to get a random selection of pupils solving each exercise, the following plan was used: Each exercise was printed on a separate sheet of paper in the manner 1 J. H. Minnick. An Investigation of Certain Abilities Fundamental to the Study of Geometry. 102 THE SCHOOL REVIEW shown on pages 107-109. The exercises were then assembled in groups of three each in such a way that every possible com- bination of the exercises was taken, any one combination occurring as often as any other. The following directions were sent to each teacher who gave the tests: 1. See that each pupil is supplied with a pencil and ruler. 1. Read to the pupils: I am going to give you some geometry exercises. In order that you may all have the same chance, I want you to start at the same time. Do not open the set of questions which you are about to receive until I give the signal to begin work by tapping on the desk. 3. Distribute the questions. 4. Have pupils fill out blanks on the cover sheets of their questions. 5. Read to the pupils: At the top of each sheet which you have received there is a geometrical figure. Below this figure there is a statement of what is given and what is to be done. When the signal to begin work is given, fold back the cover sheet, read carefully what is given and what is to be proved in one of the exercises, and then make any additional drawings that are necessary to prove the exercise. Thus, if, in the triangle ABC AC = BC [place drawing on the board] and if we are to prove that Z.A = LB, we may draw CD to the mid-point of AB. When you have completed this exercise proceed in a similar way to do the other exercises. Do not write any explanation on your papers and do not prove the exercises. All I want to know is whether you can make the correct drawings. You may do the exercises in any order you prefer. When you have completed the exercises to the best of your ability, hand me your papers. 6. Give the pupils a chance to ask questions concerning the instructions, but do not reveal the contents of the questions by your answers. 7. Note the time and then give the signal, thus: "Ready," and tap on the desk with your pencil. 8. Do not permit talking in the class until all papers have been returned to you. 9. In the case of any irregularity on the part of any pupil during the test, make a note on the cover sheet of his questions indicating the exact nature of the irregularity. 10. As the pupils hand you their papers, note the amount of time required by each, and record it on the upper right-hand corner of the cover sheet of his or her paper. PUPILS' ABILITY TO DEMONSTRATE THEOREMS 103 The tests were given in thirty high schools distributed throughout the country and ranging in size from a few hundred pupils to several thousand. Each of the eleven exercises was solved by at least seven hundred pupils. These pupils had completed either the first two books of plane geometry or all of plane geometry. TABLE I Percentage of Pupils Solving Each Exercise Correctly Number of Pupils Exercises 50 ICO 150 200 250 300 35° 400 450 500 55° 600 650 700 42 37 34 34 33 35 38 38 41 41 41 42 42 42 76 70 7i 72 75 76 77 77 77 79 81 81 82 82 20 28 33 37 42 45 49 46 46 48 49 49 49 49 42 42 38 38 37 37 37 38 36 35 3° 29 28 28 68 62 63 64 64 66 67 68 66 66 66 66 67 68 28 20 22 26 28 30 32 33 35 34 32 32 3i 3i 68 60 60 64 60 61 64 64 62 60 59 58 58 57 14 12 l 3 '5 12 J 5 14 H 14 14 13 13 40 41 41 44 45 45 45 43 44 42 42 41 40 39 16 24 40 46 43 41 36 32 28 25 23 22 19 20 46 47 36 36 36 37 38 36 37 36 36 35 34 34 In order to secure uniformity of marking, all papers were marked by the author. Any drawing which made a proof possible was usually counted as a correct solution. If, how- ever, in addition to the correct lines, unnecessary lines were drawn, the solution was marked as incorrect. Also, if the proof resulting from a drawing seemed to be too difficult for high-school pupils, those pupils making the drawing were asked to complete the proof. If the majority could not do so, the drawing was not accepted as a correct solution. A careful io 4 THE SCHOOL REVIEW record of all acceptable drawings has been kept and will be included in the "Directions for Scoring Papers 1 ". Table I gives the percentage of pupils solving each exercise correctly. The table is arranged in a cumulative way, each new line including the data for fifty additional pupils. Thus of the first fifty pupils solving Exercise i, 42 per cent got it correct; of the first hundred solving Exercise i, 37 per cent got it correct, etc. The table shows that the number of pupils tested has not been sufficient to eliminate completely individ- ual variation. However, the percentages are fairly constant, and it is not probable that the addition of data from more pupils would vary the order of difficulty or seriously change the weighting of the exercises. Very poor A M Very good Fig. 1 In weighting the exercises we have assumed that the distribution of pupils according to the ability in question will result in the normal frequency-curve as shown in Fig. 1. The direction from left to right on CD will be considered positive. The line MN represents the median. If AP is drawn so that AMNP is one-fourth the entire surface under the curve, then AM is known as the P. E. (possible error) . Reference to Table I shows that Exercise 2 is the easiest of the eleven exercises used. The score for this exercise is 82. It is, therefore, 82 — 50, or 32 per cent too easy for the median pupil. Converting this difference into P. E. 2 values, we find that Exercise 2 falls at — 1.357 P. E.; that is, 1.357 P. E. to the 1 Information concerning the prices of tests and directions for giving and scoring the tests may be had by addressing J. H. Minnick, College Hall, University of Pennsylvania, Philadelphia, Pennsylvania. 2 Table XIII of Trabue's Completion Test Language Scale. PUPILS' ABILITY TO DEMONSTRATE THEOREMS 105 left of the median. Since our zero-point must be arbitrary, we may select it so as to give convenient numbers for values of the various exercises. Hence we have selected —2.357 P. E. as our zero-point. This makes Exercise 2 fall at 1 P. E. above the zero-point. Table II indicates the method of determining the value of each exercise. Thus the first line indicates that 42 per cent of the pupils solved Exercise 1 correctly, that the exercise is 8 TABLE II Values Assigned to Each Exercise Exercise Per Cent Correct Difference between 50 Per Cent and Score P. E. Value Distance in P. E. above Zero-Point Value 42.O 82.O 49-3 27.9 67.7 3*3 56.7 l 3-3 39-3 20.0 34-o + 8.0 —32.O + 0.7 + 22. 1 —17.7 + 18.7 -6.7 + 36.7 + IO.7 +30.0 + I6.0 + O.299 —'•357 +0.026 +0.869 -0.681 +0.723 — 0.250 + 1.649 +0.403 + 1.248 +0.612 2.656 1 .OOO 2 383 3.226 I.676 3.080 2.IO7 4.006 2.760 3.605 2.969 27 IO 2 7 24 3 2 17 3i 21 C 6 7 8 40 28 q IO 36 3° II per cent too difficult for the median pupil and is, therefore, 0.299 P- E. above the median or 2.656 P. E. above our arbitrary zero-point, and that the value assigned to Exercise 1 is 27. In order to avoid the decimal point in the values assigned, 0.1 P. E. has been taken as the unit. The values are then obtained by moving the decimal point one place to the right in the next to the last column of the table. Fig. 2 is the linear projection of the eleven exercises. From these exercises it was necessary to select those which should constitute the scale. When the tests were given, a io6 THE SCHOOL REVIEW record was kept of the time required by each pupil to do the three exercises assigned to him. This time varied from 5 to 30 minutes. The average time spent by a pupil was about 1 8 minutes. It, therefore, seemed that five exercises were sufficient for the 15 or 20 minutes 1 which the pupils will be given to work on the final scale. An examination of Fig. 2 shows that Exerolses- 1 r.E. 1 — 2 ».«. t Fig. 2 3 P.E. 19 11 6 4 ■"I— 10 4 P.E. >l Exercises 2, 5, 3, and 6 are distributed along the scale at almost equal intervals. The interval between 6 and 10 is somewhat smaller, and that between 6 and 8 somewhat larger than the intervals by which 2, 5, 3, and 6 are separated. Exercise 10 occurs as a theorem in some texts, and the results from the schools in which the tests were given were clearly affected by this fact. Hence Exercise 8 was selected as the fifth exercise of our scale. TABLE III Order of Difficulty for the Eight Largest Schools Schools Order 1 2 3 4 5 A 2 2 2 2 2 5 2 2 5 3 5 5 5 2 3 5 3 5 3 6 3 3 5 3 6 6 6 3 6 6 6 6 8 B 8 C 8 D 8 E 8 F 8 G 8 H 8 The selection of these five exercises is also justified by the fact that two fall above, two below, and one almost at the ■This time is to be determined experimentally during the year 1919-20. PUPILS' ABILITY TO DEMONSTRATE THEOREMS 107 median. Also, if we examine Table I, we find that these exercises have been given to enough pupils to give almost constant results. The order of difficulty of these five exercises for students completing the first two books of geometry and for pupils completing all of plane geometry has been carefully examined and has been found to be identical in the two cases. Table III shows the order of difficulty for the eight largest schools. Thus for School A the array of exercises from the easiest to the most difficult is 2, 5, 3, 6, 8. Four of the eight schools have the same order of difficulty. In each of the other schools there is one and only one reversal of the order. In no case is an exercise removed more than one place from the order given by all of the schools taken together. Exercises 2, 5, 3, 6, 8, will constitute our test and will be numbered I, II, III, IV, V, respectively. The exercises in full follow: Given: The circle whose center is O, arc AB = arc BC, BM is perpendicular to AO, BN is perpendicular to OC. Make any additional drawings that are necessary to prove that BM = BN. io8 THE SCHOOL REVIEW ll -B F Given: AB is parallel to CD, and lines GE and FE meet in E. Make any additional drawings that are necessary to prove that Z2= Zi+ Z3- HI Given: Triangle ABC inscribed in a circle whose center is 0, and OD perpendicular to CB. Make any additional drawings that are necessary to prove that Zi= Z2. PUPILS' ABILITY TO DEMONSTRATE THEOREMS 109 Given: The triangle ABC, AC = BC, D is any point on AB, DE is perpendicular to BC, DG is perpendicular to AC. Make any additional drawings necessary to prove that DG + DE = AF. Given: The circle whose center is 0, chord AC = chord BD, AC and BD intersect in P, and the line OP is drawn. Make any additional drawings that are necessary to prove that Z 1 = /.1.